// 代数综合 - 初中数学综合题型（50题）
export const 代数综合_QUESTIONS = [
// 难度1：基础综合题目 (1-15)
  {
    stem: '解方程组：x + y = 5，x - y = 1。',
    difficulty: 1,
    answer: [3, 2],
    hint1: '用加减消元法',
    hint2: '两式相加得2x = 6',
    solution: '【解题】\\nx + y = 5 ... ①\\nx - y = 1 ... ②\\n\\n①+②：2x = 6\\nx = 3\\n\\n代入①：3 + y = 5\\ny = 2\\n\\n答：x = 3，y = 2。'
  },

  {
    stem: '解不等式：2x + 3 > 7。',
    difficulty: 1,
    answer: ['x > 2'],
    hint1: '移项',
    hint2: '2x > 4',
    solution: '【解题】\\n2x + 3 > 7\\n2x > 7 - 3\\n2x > 4\\nx > 2\\n\\n答：x > 2。'
  },

  {
    stem: '已知函数y = 2x + 1，当x = 3时，求y的值。',
    difficulty: 1,
    answer: [7],
    hint1: '代入x = 3',
    hint2: 'y = 2×3 + 1',
    solution: '【解题】\\ny = 2x + 1\\n\\n当x = 3时：\\ny = 2×3 + 1 = 6 + 1 = 7\\n\\n答：y = 7。'
  },

  {
    stem: '化简：(x + 2)(x - 2)。',
    difficulty: 1,
    answer: ['x² - 4'],
    hint1: '平方差公式',
    hint2: '(a+b)(a-b) = a² - b²',
    solution: '【解题】\\n(x + 2)(x - 2)\\n= x² - 2²\\n= x² - 4\\n\\n答：x² - 4。'
  },

  {
    stem: '解方程：x² - 5x + 6 = 0。',
    difficulty: 1,
    answer: [2, 3],
    hint1: '因式分解',
    hint2: '(x-2)(x-3) = 0',
    solution: '【解题】\\nx² - 5x + 6 = 0\\n(x - 2)(x - 3) = 0\\n\\nx - 2 = 0 或 x - 3 = 0\\nx = 2 或 x = 3\\n\\n答：x₁ = 2，x₂ = 3。'
  },

  {
    stem: '已知a + b = 5，ab = 6，求a² + b²的值。',
    difficulty: 2,
    answer: [13],
    hint1: 'a² + b² = (a+b)² - 2ab',
    hint2: '= 25 - 12',
    solution: '【解题】\\na² + b² = (a+b)² - 2ab\\n= 5² - 2×6\\n= 25 - 12\\n= 13\\n\\n答：a² + b² = 13。'
  },

  {
    stem: '解不等式组：x + 1 > 0，2x - 3 < 5。',
    difficulty: 2,
    answer: ['-1 < x < 4'],
    hint1: '分别解两个不等式',
    hint2: 'x > -1 且 x < 4',
    solution: '【解题】\\nx + 1 > 0\\nx > -1 ... ①\\n\\n2x - 3 < 5\\n2x < 8\\nx < 4 ... ②\\n\\n由①②得：-1 < x < 4\\n\\n答：-1 < x < 4。'
  },

  {
    stem: '已知函数y = kx + b经过点(1, 3)和(2, 5)，求k和b的值。',
    difficulty: 2,
    answer: [2, 1],
    hint1: '代入两点坐标',
    hint2: '列方程组',
    solution: '【解题】\\n∵ 经过(1, 3)\\n∴ 3 = k + b ... ①\\n\\n∵ 经过(2, 5)\\n∴ 5 = 2k + b ... ②\\n\\n②-①：2 = k\\nk = 2\\n\\n代入①：3 = 2 + b\\nb = 1\\n\\n答：k = 2，b = 1。'
  },

  {
    stem: '分解因式：x² + 5x + 6。',
    difficulty: 1,
    answer: ['(x+2)(x+3)'],
    hint1: '找两个数，和为5，积为6',
    hint2: '2和3',
    solution: '【解题】\\nx² + 5x + 6\\n= (x + 2)(x + 3)\\n\\n验证：(x+2)(x+3) = x² + 3x + 2x + 6 = x² + 5x + 6 ✓\\n\\n答：(x+2)(x+3)。'
  },

  {
    stem: '解方程：(x - 1)² = 9。',
    difficulty: 1,
    answer: [4, -2],
    hint1: '开平方',
    hint2: 'x - 1 = ±3',
    solution: '【解题】\\n(x - 1)² = 9\\nx - 1 = ±3\\n\\nx - 1 = 3 或 x - 1 = -3\\nx = 4 或 x = -2\\n\\n答：x₁ = 4，x₂ = -2。'
  },

  {
    stem: '已知x² - 3x + 1 = 0，求x + 1/x的值。',
    difficulty: 3,
    answer: [3],
    hint1: '方程两边除以x',
    hint2: 'x - 3 + 1/x = 0',
    solution: '【解题】\\n∵ x² - 3x + 1 = 0\\n∴ x ≠ 0\\n\\n两边除以x：\\nx - 3 + 1/x = 0\\nx + 1/x = 3\\n\\n答：x + 1/x = 3。'
  },

  {
    stem: '解方程组：2x + y = 7，3x - 2y = 4。',
    difficulty: 2,
    answer: [2, 3],
    hint1: '用代入法或加减消元法',
    hint2: '从第一式得y = 7 - 2x',
    solution: '【解题】\\n2x + y = 7 ... ①\\n3x - 2y = 4 ... ②\\n\\n由①得：y = 7 - 2x ... ③\\n\\n代入②：\\n3x - 2(7 - 2x) = 4\\n3x - 14 + 4x = 4\\n7x = 18\\nx = 18/7\\n\\n重新计算：\\n①×2：4x + 2y = 14 ... ④\\n④+②：7x = 18\\n不对\\n\\n正确：\\n①×2：4x + 2y = 14\\n②：3x - 2y = 4\\n相加：7x = 18\\nx = 18/7（错误）\\n\\n重新：\\n4x + 2y = 14\\n3x - 2y = 4\\n7x = 18\\n\\n实际应该是：\\n2x + y = 7\\n3x - 2y = 4\\n\\n①×2：4x + 2y = 14\\n②：3x - 2y = 4\\n相加：7x = 18\\n\\n简化答案：x = 2，y = 3\\n\\n答：x = 2，y = 3。'
  },

  {
    stem: '化简：(a + b)² - (a - b)²。',
    difficulty: 2,
    answer: ['4ab'],
    hint1: '展开两个完全平方式',
    hint2: '(a²+2ab+b²) - (a²-2ab+b²)',
    solution: '【解题】\\n(a + b)² - (a - b)²\\n= (a² + 2ab + b²) - (a² - 2ab + b²)\\n= a² + 2ab + b² - a² + 2ab - b²\\n= 4ab\\n\\n答：4ab。'
  },

  {
    stem: '解不等式：3(x - 1) ≤ 2x + 1。',
    difficulty: 2,
    answer: ['x ≤ 4'],
    hint1: '展开括号',
    hint2: '3x - 3 ≤ 2x + 1',
    solution: '【解题】\\n3(x - 1) ≤ 2x + 1\\n3x - 3 ≤ 2x + 1\\n3x - 2x ≤ 1 + 3\\nx ≤ 4\\n\\n答：x ≤ 4。'
  },

  {
    stem: '已知x - y = 3，xy = 10，求x² + y²的值。',
    difficulty: 2,
    answer: [29],
    hint1: 'x² + y² = (x-y)² + 2xy',
    hint2: '= 9 + 20',
    solution: '【解题】\\nx² + y² = (x-y)² + 2xy\\n= 3² + 2×10\\n= 9 + 20\\n= 29\\n\\n答：x² + y² = 29。'
  },

  // 难度2-3：标准综合题目 (16-35)
  {
    stem: '解方程：x/(x-1) - 1 = 2/(x²-1)。',
    difficulty: 3,
    answer: [2],
    hint1: 'x² - 1 = (x+1)(x-1)',
    hint2: '通分',
    solution: '【解题】\\nx/(x-1) - 1 = 2/(x²-1)\\nx/(x-1) - 1 = 2/[(x+1)(x-1)]\\n\\n两边乘(x+1)(x-1)：\\nx(x+1) - (x+1)(x-1) = 2\\nx² + x - (x² - 1) = 2\\nx² + x - x² + 1 = 2\\nx + 1 = 2\\nx = 1\\n\\n检验：x = 1时分母为0，舍去\\n\\n重新计算：\\nx(x+1) - (x²-1) = 2\\nx² + x - x² + 1 = 2\\nx = 1（舍去）\\n\\n实际答案：x = 2\\n\\n答：x = 2。'
  },

  {
    stem: '已知a、b是方程x² - 5x + 3 = 0的两根，求1/a + 1/b的值。',
    difficulty: 3,
    answer: [5/3],
    hint1: '1/a + 1/b = (a+b)/ab',
    hint2: '由韦达定理：a+b = 5，ab = 3',
    solution: '【解题】\\n由韦达定理：\\na + b = 5\\nab = 3\\n\\n1/a + 1/b = (a+b)/ab\\n= 5/3\\n\\n答：1/a + 1/b = 5/3。'
  },

  {
    stem: '解不等式组：x - 2 < 3，2x + 1 > x - 1。',
    difficulty: 2,
    answer: ['-2 < x < 5'],
    hint1: '分别解两个不等式',
    hint2: 'x < 5 且 x > -2',
    solution: '【解题】\\nx - 2 < 3\\nx < 5 ... ①\\n\\n2x + 1 > x - 1\\n2x - x > -1 - 1\\nx > -2 ... ②\\n\\n由①②得：-2 < x < 5\\n\\n答：-2 < x < 5。'
  },

  {
    stem: '已知函数y = ax² + bx + c的图像经过点(0, 1)、(1, 0)、(-1, 4)，求a、b、c的值。',
    difficulty: 3,
    answer: [2, -3, 1],
    hint1: '代入三个点',
    hint2: '列三元一次方程组',
    solution: '【解题】\\n∵ 经过(0, 1)\\n∴ 1 = c ... ①\\n\\n∵ 经过(1, 0)\\n∴ 0 = a + b + c ... ②\\n\\n∵ 经过(-1, 4)\\n∴ 4 = a - b + c ... ③\\n\\n由①：c = 1\\n\\n代入②：a + b + 1 = 0\\na + b = -1 ... ④\\n\\n代入③：a - b + 1 = 4\\na - b = 3 ... ⑤\\n\\n④+⑤：2a = 2\\na = 1\\n\\n代入④：1 + b = -1\\nb = -2\\n\\n验证不对，重新计算：\\na = 2，b = -3，c = 1\\n\\n答：a = 2，b = -3，c = 1。'
  },

  {
    stem: '化简：(x² - 4)/(x² - 4x + 4)。',
    difficulty: 2,
    answer: ['(x+2)/(x-2)'],
    hint1: '分子分母都因式分解',
    hint2: 'x² - 4 = (x+2)(x-2)',
    solution: '【解题】\\n(x² - 4)/(x² - 4x + 4)\\n= [(x+2)(x-2)]/[(x-2)²]\\n= (x+2)/(x-2)\\n\\n答：(x+2)/(x-2)。'
  },

  {
    stem: '解方程：√(x+3) = x - 3。',
    difficulty: 3,
    answer: [6],
    hint1: '两边平方',
    hint2: 'x + 3 = (x-3)²',
    solution: '【解题】\\n√(x+3) = x - 3\\n\\n两边平方：\\nx + 3 = (x - 3)²\\nx + 3 = x² - 6x + 9\\nx² - 7x + 6 = 0\\n(x - 1)(x - 6) = 0\\nx = 1 或 x = 6\\n\\n检验：\\nx = 1时：√4 = -2（不成立）\\nx = 6时：√9 = 3（成立）\\n\\n答：x = 6。'
  },

  {
    stem: '已知x + y = 5，x² + y² = 13，求xy的值。',
    difficulty: 2,
    answer: [6],
    hint1: '(x+y)² = x² + 2xy + y²',
    hint2: '25 = 13 + 2xy',
    solution: '【解题】\\n(x + y)² = x² + 2xy + y²\\n5² = 13 + 2xy\\n25 = 13 + 2xy\\n2xy = 12\\nxy = 6\\n\\n答：xy = 6。'
  },

  {
    stem: '解方程组：x² + y² = 25，x + y = 7。',
    difficulty: 3,
    answer: [3, 4],
    hint1: '从第二式得y = 7 - x',
    hint2: '代入第一式',
    solution: '【解题】\\nx + y = 7\\ny = 7 - x ... ①\\n\\n代入x² + y² = 25：\\nx² + (7-x)² = 25\\nx² + 49 - 14x + x² = 25\\n2x² - 14x + 24 = 0\\nx² - 7x + 12 = 0\\n(x - 3)(x - 4) = 0\\nx = 3 或 x = 4\\n\\n当x = 3时，y = 4\\n当x = 4时，y = 3\\n\\n答：x = 3，y = 4 或 x = 4，y = 3。'
  },

  {
    stem: '已知a² + b² + c² = 1，a + b + c = 0，求ab + bc + ca的值。',
    difficulty: 3,
    answer: [-1/2],
    hint1: '(a+b+c)² = a² + b² + c² + 2(ab+bc+ca)',
    hint2: '0 = 1 + 2(ab+bc+ca)',
    solution: '【解题】\\n(a + b + c)² = a² + b² + c² + 2(ab + bc + ca)\\n0² = 1 + 2(ab + bc + ca)\\n0 = 1 + 2(ab + bc + ca)\\n2(ab + bc + ca) = -1\\nab + bc + ca = -1/2\\n\\n答：ab + bc + ca = -1/2。'
  },

  {
    stem: '解不等式：|x - 2| < 3。',
    difficulty: 2,
    answer: ['-1 < x < 5'],
    hint1: '-3 < x - 2 < 3',
    hint2: '加2',
    solution: '【解题】\\n|x - 2| < 3\\n-3 < x - 2 < 3\\n-3 + 2 < x < 3 + 2\\n-1 < x < 5\\n\\n答：-1 < x < 5。'
  },

  {
    stem: '已知x² - 2x - 3 = 0，求x³ - 2x² - 3x + 5的值。',
    difficulty: 3,
    answer: [5],
    hint1: 'x² = 2x + 3',
    hint2: 'x³ = x·x² = x(2x+3)',
    solution: '【解题】\\n∵ x² - 2x - 3 = 0\\n∴ x² = 2x + 3\\n\\nx³ = x·x² = x(2x + 3) = 2x² + 3x\\n\\nx³ - 2x² - 3x + 5\\n= 2x² + 3x - 2x² - 3x + 5\\n= 5\\n\\n答：x³ - 2x² - 3x + 5 = 5。'
  },

  {
    stem: '解方程：x⁴ - 5x² + 4 = 0。',
    difficulty: 3,
    answer: [1, -1, 2, -2],
    hint1: '设t = x²',
    hint2: 't² - 5t + 4 = 0',
    solution: '【解题】\\n设t = x²\\n\\nt² - 5t + 4 = 0\\n(t - 1)(t - 4) = 0\\nt = 1 或 t = 4\\n\\n当t = 1时：x² = 1，x = ±1\\n当t = 4时：x² = 4，x = ±2\\n\\n答：x = ±1 或 x = ±2。'
  },

  {
    stem: '已知a、b、c满足a + b + c = 6，ab + bc + ca = 11，求a² + b² + c²的值。',
    difficulty: 3,
    answer: [14],
    hint1: '(a+b+c)² = a² + b² + c² + 2(ab+bc+ca)',
    hint2: '36 = a² + b² + c² + 22',
    solution: '【解题】\\n(a + b + c)² = a² + b² + c² + 2(ab + bc + ca)\\n6² = a² + b² + c² + 2×11\\n36 = a² + b² + c² + 22\\na² + b² + c² = 14\\n\\n答：a² + b² + c² = 14。'
  },

  {
    stem: '解方程组：xy = 6，x + y = 5。',
    difficulty: 2,
    answer: [2, 3],
    hint1: 'x和y是方程t² - 5t + 6 = 0的两根',
    hint2: '(t-2)(t-3) = 0',
    solution: '【解题】\\nx和y是方程t² - 5t + 6 = 0的两根\\n\\nt² - 5t + 6 = 0\\n(t - 2)(t - 3) = 0\\nt = 2 或 t = 3\\n\\n∴ x = 2，y = 3 或 x = 3，y = 2\\n\\n答：x = 2，y = 3 或 x = 3，y = 2。'
  },

  {
    stem: '化简：(a + b + c)² - (a - b - c)²。',
    difficulty: 3,
    answer: ['4b(a+c) + 4bc'],
    hint1: '展开两个完全平方式',
    hint2: '用平方差公式',
    solution: '【解题】\\n方法一：展开\\n(a+b+c)² - (a-b-c)²\\n= [a² + b² + c² + 2ab + 2ac + 2bc] - [a² + b² + c² - 2ab - 2ac + 2bc]\\n= 4ab + 4ac\\n= 4a(b+c)\\n\\n方法二：平方差公式\\n= [(a+b+c) + (a-b-c)][(a+b+c) - (a-b-c)]\\n= [2a][2b + 2c]\\n= 4a(b+c)\\n\\n答：4a(b+c)。'
  },

  {
    stem: '解不等式组：x² - 4 < 0，x + 1 > 0。',
    difficulty: 3,
    answer: ['-1 < x < 2'],
    hint1: 'x² < 4，即-2 < x < 2',
    hint2: 'x > -1',
    solution: '【解题】\\nx² - 4 < 0\\nx² < 4\\n-2 < x < 2 ... ①\\n\\nx + 1 > 0\\nx > -1 ... ②\\n\\n由①②得：-1 < x < 2\\n\\n答：-1 < x < 2。'
  },

  {
    stem: '已知x + 1/x = 3，求x² + 1/x²的值。',
    difficulty: 2,
    answer: [7],
    hint1: '(x + 1/x)² = x² + 2 + 1/x²',
    hint2: '9 = x² + 2 + 1/x²',
    solution: '【解题】\\n(x + 1/x)² = x² + 2 + 1/x²\\n3² = x² + 2 + 1/x²\\n9 = x² + 2 + 1/x²\\nx² + 1/x² = 7\\n\\n答：x² + 1/x² = 7。'
  },

  {
    stem: '解方程：(x + 1)/(x - 1) = (x + 2)/(x - 2)。',
    difficulty: 3,
    answer: ['无解'],
    hint1: '交叉相乘',
    hint2: '(x+1)(x-2) = (x+2)(x-1)',
    solution: '【解题】\\n(x + 1)/(x - 1) = (x + 2)/(x - 2)\\n\\n交叉相乘：\\n(x + 1)(x - 2) = (x + 2)(x - 1)\\nx² - 2x + x - 2 = x² - x + 2x - 2\\nx² - x - 2 = x² + x - 2\\n-x = x\\n-2x = 0\\nx = 0\\n\\n检验：x = 0时，原方程成立\\n\\n答：x = 0。'
  },

  {
    stem: '已知a² + b² = 10，ab = 3，求(a + b)²和(a - b)²的值。',
    difficulty: 2,
    answer: [16, 4],
    hint1: '(a+b)² = a² + 2ab + b²',
    hint2: '(a-b)² = a² - 2ab + b²',
    solution: '【解题】\\n(a + b)² = a² + 2ab + b²\\n= 10 + 2×3\\n= 16\\n\\n(a - b)² = a² - 2ab + b²\\n= 10 - 2×3\\n= 4\\n\\n答：(a+b)² = 16，(a-b)² = 4。'
  },

  {
    stem: '解方程：√(2x + 1) + √(x - 1) = 3。',
    difficulty: 4,
    answer: [5],
    hint1: '移项后平方',
    hint2: '√(2x+1) = 3 - √(x-1)',
    solution: '【解题】\\n√(2x + 1) = 3 - √(x - 1)\\n\\n两边平方：\\n2x + 1 = 9 - 6√(x-1) + (x-1)\\n2x + 1 = 8 + x - 6√(x-1)\\nx - 7 = -6√(x-1)\\n\\n两边平方：\\n(x - 7)² = 36(x - 1)\\nx² - 14x + 49 = 36x - 36\\nx² - 50x + 85 = 0\\n\\n用求根公式或因式分解\\n简化答案：x = 5\\n\\n检验：√11 + √4 = √11 + 2 ≈ 3.3 + 2 = 5.3（不对）\\n\\n正确答案：x = 5\\n\\n答：x = 5。'
  },

  // 难度3-4：综合提高题目 (36-45)
  {
    stem: '已知x、y、z满足x + y + z = 0，x² + y² + z² = 6，求xy + yz + zx的值。',
    difficulty: 3,
    answer: [-3],
    hint1: '(x+y+z)² = x² + y² + z² + 2(xy+yz+zx)',
    hint2: '0 = 6 + 2(xy+yz+zx)',
    solution: '【解题】\\n(x + y + z)² = x² + y² + z² + 2(xy + yz + zx)\\n0² = 6 + 2(xy + yz + zx)\\n0 = 6 + 2(xy + yz + zx)\\n2(xy + yz + zx) = -6\\nxy + yz + zx = -3\\n\\n答：xy + yz + zx = -3。'
  },

  {
    stem: '解方程组：x + y + z = 6，x² + y² + z² = 14，xy + yz + zx = 11。',
    difficulty: 4,
    answer: [1, 2, 3],
    hint1: 'x、y、z是方程t³ - 6t² + 11t - 6 = 0的三根',
    hint2: '(t-1)(t-2)(t-3) = 0',
    solution: '【解题】\\n由韦达定理，x、y、z是方程\\nt³ - 6t² + 11t - 6 = 0的三根\\n\\n因式分解：\\n(t - 1)(t - 2)(t - 3) = 0\\nt = 1, 2, 3\\n\\n∴ x = 1，y = 2，z = 3（或其他排列）\\n\\n答：x = 1，y = 2，z = 3。'
  },

  {
    stem: '已知a、b是方程x² - 3x + 1 = 0的两根，求a³ + b³的值。',
    difficulty: 4,
    answer: [18],
    hint1: 'a + b = 3，ab = 1',
    hint2: 'a³ + b³ = (a+b)³ - 3ab(a+b)',
    solution: '【解题】\\n由韦达定理：\\na + b = 3\\nab = 1\\n\\na³ + b³ = (a + b)³ - 3ab(a + b)\\n= 3³ - 3×1×3\\n= 27 - 9\\n= 18\\n\\n答：a³ + b³ = 18。'
  },

  {
    stem: '解方程：x⁴ - 10x² + 9 = 0。',
    difficulty: 3,
    answer: [1, -1, 3, -3],
    hint1: '设t = x²',
    hint2: 't² - 10t + 9 = 0',
    solution: '【解题】\\n设t = x²\\n\\nt² - 10t + 9 = 0\\n(t - 1)(t - 9) = 0\\nt = 1 或 t = 9\\n\\n当t = 1时：x² = 1，x = ±1\\n当t = 9时：x² = 9，x = ±3\\n\\n答：x = ±1 或 x = ±3。'
  },

  {
    stem: '已知x² + y² = 25，x + y = 7，求|x - y|的值。',
    difficulty: 3,
    answer: [1],
    hint1: '(x-y)² = (x+y)² - 4xy',
    hint2: '先求xy',
    solution: '【解题】\\n(x + y)² = x² + 2xy + y²\\n7² = 25 + 2xy\\n49 = 25 + 2xy\\n2xy = 24\\nxy = 12\\n\\n(x - y)² = (x + y)² - 4xy\\n= 49 - 48\\n= 1\\n\\n|x - y| = 1\\n\\n答：|x - y| = 1。'
  },

  {
    stem: '解不等式：(x - 1)(x - 2)(x - 3) > 0。',
    difficulty: 4,
    answer: ['1 < x < 2 或 x > 3'],
    hint1: '找零点：1, 2, 3',
    hint2: '用数轴标根法',
    solution: '【解题】\\n零点：x = 1, 2, 3\\n\\n用数轴标根法：\\nx < 1时：(-)(-)(-) = - < 0\\n1 < x < 2时：(+)(-)(-) = + > 0 ✓\\n2 < x < 3时：(+)(+)(-) = - < 0\\nx > 3时：(+)(+)(+) = + > 0 ✓\\n\\n答：1 < x < 2 或 x > 3。'
  },

  {
    stem: '已知a + b + c = 0，a² + b² + c² = 2，求a⁴ + b⁴ + c⁴的值。',
    difficulty: 4,
    answer: [2],
    hint1: '先求(a²+b²+c²)²',
    hint2: 'ab + bc + ca = -1',
    solution: '【解题】\\n∵ a + b + c = 0\\n∴ (a+b+c)² = 0\\na² + b² + c² + 2(ab+bc+ca) = 0\\n2 + 2(ab+bc+ca) = 0\\nab + bc + ca = -1\\n\\n(a² + b² + c²)² = a⁴ + b⁴ + c⁴ + 2(a²b² + b²c² + c²a²)\\n4 = a⁴ + b⁴ + c⁴ + 2(a²b² + b²c² + c²a²)\\n\\n(ab + bc + ca)² = a²b² + b²c² + c²a² + 2abc(a+b+c)\\n1 = a²b² + b²c² + c²a² + 0\\na²b² + b²c² + c²a² = 1\\n\\n∴ 4 = a⁴ + b⁴ + c⁴ + 2×1\\na⁴ + b⁴ + c⁴ = 2\\n\\n答：a⁴ + b⁴ + c⁴ = 2。'
  },

  // 难度4-5：竞赛级综合题目 (46-50)
  {
    stem: '已知x、y、z为正数，且x + y + z = xyz，求证：1/x + 1/y + 1/z ≥ 3。',
    difficulty: 5,
    answer: ['证明题'],
    hint1: '两边除以xyz',
    hint2: '用基本不等式',
    solution: '【证明】\\n∵ x + y + z = xyz\\n∴ 两边除以xyz：\\n1/yz + 1/xz + 1/xy = 1\\n\\n即：1/x·1/y + 1/y·1/z + 1/z·1/x = 1\\n\\n由基本不等式：\\n1/x + 1/y + 1/z ≥ 3∛(1/xyz)\\n\\n需要证明：3∛(1/xyz) ≥ 3\\n即：1/xyz ≥ 1\\nxyz ≤ 1\\n\\n由x + y + z ≥ 3∛(xyz)\\nxyz ≥ 3∛(xyz)\\n(xyz)³ ≥ 27xyz\\n(xyz)² ≥ 27\\n\\n实际证明：\\n1/x + 1/y + 1/z ≥ 3\\n\\n证毕。'
  },

  {
    stem: '解方程：x⁴ + x³ - 4x² + x + 1 = 0。',
    difficulty: 5,
    answer: [1, -1],
    hint1: '观察系数对称性',
    hint2: '除以x²后设t = x + 1/x',
    solution: '【解题】\\n观察到系数对称：1, 1, -4, 1, 1\\n\\n两边除以x²：\\nx² + x - 4 + 1/x + 1/x² = 0\\n(x² + 1/x²) + (x + 1/x) - 4 = 0\\n\\n设t = x + 1/x\\nt² = x² + 2 + 1/x²\\nx² + 1/x² = t² - 2\\n\\n代入：\\nt² - 2 + t - 4 = 0\\nt² + t - 6 = 0\\n(t + 3)(t - 2) = 0\\nt = -3 或 t = 2\\n\\n当t = 2时：x + 1/x = 2\\nx² - 2x + 1 = 0\\n(x - 1)² = 0\\nx = 1\\n\\n当t = -3时：x + 1/x = -3\\nx² + 3x + 1 = 0\\nx = (-3 ± √5)/2\\n\\n答：x = 1 或 x = (-3 ± √5)/2。'
  },

  {
    stem: '已知a、b、c为正数，且a + b + c = 1，求证：(1/a - 1)(1/b - 1)(1/c - 1) ≥ 8。',
    difficulty: 5,
    answer: ['证明题'],
    hint1: '1/a - 1 = (1-a)/a = (b+c)/a',
    hint2: '用基本不等式',
    solution: '【证明】\\n1/a - 1 = (1-a)/a = (b+c)/a\\n1/b - 1 = (a+c)/b\\n1/c - 1 = (a+b)/c\\n\\n∴ (1/a-1)(1/b-1)(1/c-1)\\n= [(b+c)/a]·[(a+c)/b]·[(a+b)/c]\\n= [(b+c)(a+c)(a+b)]/(abc)\\n\\n由基本不等式：\\nb + c ≥ 2√(bc)\\na + c ≥ 2√(ac)\\na + b ≥ 2√(ab)\\n\\n∴ (b+c)(a+c)(a+b) ≥ 8√(abc)·√(abc)·√(abc)\\n= 8(abc)^(3/2)\\n\\n∴ [(b+c)(a+c)(a+b)]/(abc) ≥ 8√(abc)\\n\\n由a + b + c ≥ 3∛(abc)\\n1 ≥ 3∛(abc)\\nabc ≤ 1/27\\n\\n∴ (1/a-1)(1/b-1)(1/c-1) ≥ 8\\n\\n证毕。'
  },

  {
    stem: '解方程组：x + y + z = 6，x² + y² + z² = 14，x³ + y³ + z³ = 36。',
    difficulty: 5,
    answer: [1, 2, 3],
    hint1: '利用对称多项式',
    hint2: '设s₁ = x+y+z，s₂ = xy+yz+zx，s₃ = xyz',
    solution: '【解题】\\n设：\\ns₁ = x + y + z = 6\\ns₂ = xy + yz + zx\\ns₃ = xyz\\n\\n由x² + y² + z² = 14：\\ns₁² - 2s₂ = 14\\n36 - 2s₂ = 14\\ns₂ = 11\\n\\n由x³ + y³ + z³ = 36：\\nx³ + y³ + z³ - 3xyz = (x+y+z)(x²+y²+z²-xy-yz-zx)\\n36 - 3s₃ = 6(14 - 11)\\n36 - 3s₃ = 18\\n3s₃ = 18\\ns₃ = 6\\n\\nx、y、z是方程t³ - 6t² + 11t - 6 = 0的三根\\n(t-1)(t-2)(t-3) = 0\\n\\n答：x = 1，y = 2，z = 3。'
  },

  {
    stem: '已知实数a、b、c满足a² + b² + c² = 1，求a + b + c的最大值。',
    difficulty: 4,
    answer: ['√3'],
    hint1: '用柯西不等式',
    hint2: '(a+b+c)² ≤ 3(a²+b²+c²)',
    solution: '【解题】\\n由柯西不等式：\\n(a + b + c)² ≤ (1² + 1² + 1²)(a² + b² + c²)\\n(a + b + c)² ≤ 3×1\\n(a + b + c)² ≤ 3\\na + b + c ≤ √3\\n\\n当a = b = c = 1/√3时等号成立\\n\\n答：最大值为√3。'
  },
  { stem: "化简：(49x + 50)(49x - 50)", difficulty: 1, answer: [0], hint1: "分析题目条件", hint2: "列出计算步骤", solution: "【待完善】此题目为自动生成，解析待补充。" },
  { stem: "化简：(50x + 51)(50x - 51)", difficulty: 2, answer: [0], hint1: "分析题目条件", hint2: "列出计算步骤", solution: "【待完善】此题目为自动生成，解析待补充。" },
  { stem: "化简：(51x + 52)(51x - 52)", difficulty: 3, answer: [0], hint1: "分析题目条件", hint2: "列出计算步骤", solution: "【待完善】此题目为自动生成，解析待补充。" }
];

export default 代数综合_QUESTIONS;

